Expression of type ExprTuple

from the theory of proveit.physics.quantum.algebra

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, ExprTuple, Lambda, M, i, n
from proveit.core_expr_types import a_1_to_n, a_i, v_1_to_n, v_i
from proveit.linear_algebra import Adj, Hspace, LinMap, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, Iff, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = Adj(M)
expr = ExprTuple(Lambda(M, Conditional(Iff(Equals(Qmult(sub_expr1, M), Qmult(M, sub_expr1)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, Qmult(Ket(v_i), Bra(v_i))), domain = Interval(one, n))), domain = Complex).with_wrapping(), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping()).with_wrapping_at(2), InSet(M, LinMap(Hspace, Hspace)))))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(M \mapsto \left\{\begin{array}{c} \begin{array}{l} \left(\left(M^{\dagger} \thinspace M\right) = \left(M \thinspace M^{\dagger}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\\
\left(M = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right] \end{array} \end{array} \textrm{ if } M \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)\right..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameter: 52 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operands: 6
4	Operation	operator: 73 operands: 7
5	Literal
6	ExprTuple	8, 9
7	ExprTuple	52, 10
8	Operation	operator: 46 operands: 11
9	Operation	operator: 29 operand: 17
10	Operation	operator: 13 operands: 14
11	ExprTuple	15, 16
12	ExprTuple	17
13	Literal
14	ExprTuple	44, 44
15	Operation	operator: 81 operands: 18
16	Operation	operator: 81 operands: 19
17	Lambda	parameters: 20 body: 21
18	ExprTuple	22, 52
19	ExprTuple	52, 22
20	ExprTuple	23
21	Conditional	value: 24 condition: 25
22	Operation	operator: 26 operand: 52
23	ExprRange	lambda_map: 28 start_index: 87 end_index: 88
24	Operation	operator: 29 operand: 32
25	Operation	operator: 73 operands: 31
26	Literal
27	ExprTuple	52
28	Lambda	parameter: 79 body: 70
29	Literal
30	ExprTuple	32
31	ExprTuple	33, 34
32	Lambda	parameters: 35 body: 36
33	Operation	operator: 50 operand: 43
34	Operation	operator: 38 operand: 44
35	ExprTuple	40
36	Conditional	value: 41 condition: 42
37	ExprTuple	43
38	Literal
39	ExprTuple	44
40	ExprRange	lambda_map: 45 start_index: 87 end_index: 88
41	Operation	operator: 46 operands: 47
42	Operation	operator: 48 operands: 49
43	Operation	operator: 50 operands: 51
44	Variable
45	Lambda	parameter: 79 body: 68
46	Literal
47	ExprTuple	52, 53
48	Literal
49	ExprTuple	54
50	Literal
51	ExprTuple	55
52	Variable
53	Operation	operator: 56 operand: 60
54	ExprRange	lambda_map: 58 start_index: 87 end_index: 88
55	ExprRange	lambda_map: 59 start_index: 87 end_index: 88
56	Literal
57	ExprTuple	60
58	Lambda	parameter: 79 body: 61
59	Lambda	parameter: 79 body: 62
60	Lambda	parameter: 95 body: 63
61	Operation	operator: 73 operands: 64
62	Operation	operator: 89 operand: 70
63	Conditional	value: 66 condition: 67
64	ExprTuple	68, 69
65	ExprTuple	70
66	Operation	operator: 71 operands: 72
67	Operation	operator: 73 operands: 74
68	IndexedVar	variable: 80 index: 79
69	Literal
70	IndexedVar	variable: 93 index: 79
71	Literal
72	ExprTuple	76, 77
73	Literal
74	ExprTuple	95, 78
75	ExprTuple	79
76	IndexedVar	variable: 80 index: 95
77	Operation	operator: 81 operands: 82
78	Operation	operator: 83 operands: 84
79	Variable
80	Variable
81	Literal
82	ExprTuple	85, 86
83	Literal
84	ExprTuple	87, 88
85	Operation	operator: 89 operand: 92
86	Operation	operator: 90 operand: 92
87	Literal
88	Variable
89	Literal
90	Literal
91	ExprTuple	92
92	IndexedVar	variable: 93 index: 95
93	Variable
94	ExprTuple	95
95	Variable